Translation:I will have received all the computers next week.
Yes, I know "Between now and next week" is another way of saying "By next week", but this is exactly not what the Danish sentence implies. Instead, it implies "next week" only, and not "between now and next week". If I were to tell you that I would pay you a visit next week, surely you wouldn't expect me before, as "between now and next week" would imply. (I too am a native English speaker.)
It just occurred to me that you could in fact say “All the computers will be sent next week.” That is, it could happen that they all get sent out in one batch, and that it’s going to happen next week. But that’s not what the Danish means here.
Future perfect is different than simple future. They have two (or more) different meanings. This is future perfect, and it means that something will have happened by or before a future point in time.
It works the same in English and Danish, by the way. And I suspect the Danish word 'inden' is probably a more appropriate preposition to use here than 'i'.